On unitary representations of nilpotent gauge groups (Q1416853)

The authors study representations of some groups of smooth maps \(C^\infty(X, G)\), where \(X\) is the unit sphere in \(\mathbb{R}^m\) and \(G\) is a nilpotent Lie group of Heisenberg type. They find families of irreducible unitary representations of \(C^\infty(X,G)\), which are continuous tensor product representations [\textit{S. Albeverio}, \textit{R. Høegh-Krohn}, \textit{J. A. Marion}, \textit{D. H. Testard} and \textit{B. S. Torrésani}, Noncommutative distributions: unitary representation of gauge groups and algebras (Pure and Applied Mathematics 175, Marcel Dekker, New York) (1993; Zbl 0791.22010); \textit{A. Guichardet}, Symmetric Hilbert spaces and related topics. Infinitely divisible positive definite functions, continuous products and tensor products, Gaussian and Poissonian stochastic processes (Lecture Notes in Mathematics 261, Springer-Verlag, Berlin, Heidelberg, New York) (1972; Zbl 0265.43008)]. They transform these representations by isometries of \(X\) or automorphisms of \(G\): let \(\pi\) be a unitary representation of \(C^\infty(X, G)\), \(\phi\) a diffeomorphism of \(X\) and \(a\) an automorphism of \(G\), the corresponding transformed representations are \(\pi_\phi(g)= \pi(g\circ\phi)\) and \(\pi^a(g)= \pi(a\circ g)\). They construct a generalized Fock representation and its transformed representations, study their mutual equivalence by restricting them to the center, show their special properties to answer various questions and conjectures about the unitary duals of infinite-dimensional groups.